SCHWEITZER 1976 Problem 9

Let $D$ be a convex subset of the $n$ -dimensional space, and suppose that $D'$ is obtained from $D$ by applying a positive central dilatation and then a translation. Suppose also that the sum of the volumes of $D$ and $D'$ is $1$ , and $D \cap D'\not= \emptyset .$ Determine the supremum of the volume of the convex hull of $D \cup D'$ taken for all such pairs of sets $D,D'$ .

L. Fejes-Toth, E. Makai